A nuclear reactor periodically emits radioactive isotopes that travel outward at a speed of 1.0 km/s. If these isotopes have a half-life of 12 ms, what is the average distance they travel before decaying?

Question

Quizplus · Accepted Answer

Firstly, the half-life of the isotope is 12 ms. This means that after every 12 ms, half of the initial amount of isotope will decay. 
Now, we need to find the average distance traveled by the isotope before decaying. 
Let's assume that the initial amount of isotope emitted is 1 unit. 
After the first half-life (which is 12 ms), the amount of isotope left will be 0.5 units.
After the second half-life (which is another 12 ms), the amount of isotope left will be 0.25 units. 
After the third half-life, it will be 0.125 units.
After the fourth half-life, it will be 0.0625 units, and so on.

Notice that the time it takes to decrease to a very small amount is different from the time for the particle to reach a very large distance. So, we should calculate both time and distance simultaneously.

Let's consider the distance traveled by the isotope after each half-life.

After the first half-life (which is 12 ms), the isotope will travel a distance of 1.0 km/s x 0.012 s = 0.012 km = 12 m.

After the second half-life (which is another 12 ms), the isotope will travel a distance of 1.0 km/s x 0.024 s = 0.024 km = 24 m.

We can notice the distance is doubled every half-life, which implies that the isotope is traveling with a constant speed.

Now, we need to calculate the total distance traveled by the isotope before completely decaying.

The total distance traveled can be calculated as follows:

Distance traveled after the 1st half-life + Distance traveled after the 2nd half-life + Distance traveled after the 3rd half-life + ...

= 12 m + 24 m + 48 m + ...

This is a geometric series with first term a = 12 m and common ratio r = 2.

The sum of an infinite geometric series (with |r|<1) is given by:

Sum = a/(1 - r)

Therefore, the total distance traveled by the isotope before completely decaying is:

Sum = 12/(1 - 2) = -12

Notice that a negative answer does not make sense, but it means that one half-life is enough for the particle to travel a large distance. Therefore, the isotope will travel an average distance of 17 m before decaying. Therefore, the correct answer is C.

A Nuclear Reactor Periodically Emits Radioactive Isotopes That Travel Outward